Fund Quantum Mechanics Lect & HW Solutions 44

Fund Quantum - one E 111 then in error Answer The energies are E x 1 = ¯ h 2 π 2 2 mℓ 2 x E y 1 = ¯ h 2 π 2 2 mℓ 2 y E z 1 = ¯ h 2 π 2 2

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26 CHAPTER 3. BASIC IDEAS OF QUANTUM MECHANICS Answer: Almost every word in the above story is a gross misstatement of what nature really is like when examined on quantum scales. A particle does not have a position, so phrases like “hits an end”, “reFect backward”, and “keep moving” are truly meaningless. On macroscopic scales a particle may have an relatively precisely de±ned position, but that is only because there is uncertainty in energy. If you could bring a macroscopic particle truly into a single energy eigenstate, it too would have no position. And the smallest thing you might do to ±gure out where it is would kick it out of that single energy state. 3.5.8 Three-dimensional solution 3.5.8.1 Solution pipeg-a Question: If the cross section dimensions y and z are one tenth the size of the pipe length, how much bigger are the energies E y 1 and E z 1 compared to E x 1 ? So, by what percentage is the one-dimensional ground state energy E x 1 as an approximation to the three-dimensional
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Unformatted text preview: one, E 111 , then in error? Answer: The energies are E x 1 = ¯ h 2 π 2 2 mℓ 2 x E y 1 = ¯ h 2 π 2 2 mℓ 2 y E z 1 = ¯ h 2 π 2 2 mℓ 2 z . If ℓ y and ℓ z are ten times smaller than ℓ x then E y 1 and E z 1 are each 100 times larger than E x 1 . So the one-dimensional ground state energy E x 1 is smaller than the true ground state energy E 111 = E x 1 + E y 1 + E z 1 by a factor 201. Which means it is o² by 20,000%. 3.5.8.2 Solution pipeg-b Question: At what ratio of ℓ y /ℓ x does the energy E 121 become higher than the energy E 311 ? Answer: Using the given expression for E n x n y n z , E n x n y n z = n 2 x ¯ h 2 π 2 2 mℓ 2 x + n 2 y ¯ h 2 π 2 2 mℓ 2 y + n 2 z ¯ h 2 π 2 2 mℓ 2 z , E 121 = E 311 when ¯ h 2 π 2 2 mℓ 2 x + 4¯ h 2 π 2 2 mℓ 2 y + ¯ h 2 π 2 2 mℓ 2 z = 9¯ h 2 π 2 2 mℓ 2 x + ¯ h 2 π 2 2 mℓ 2 y + ¯ h 2 π 2 2 mℓ 2 z...
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This note was uploaded on 01/06/2012 for the course PHY 3604 taught by Professor Dr.danielarenas during the Fall '11 term at UNF.

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